Seeley suggests dancing is regulated by efficiency: \(\frac{gain-cost}{cost}\).
- Gain is determined by the nectar quality of the flower, \(q\).
- Cost is determined by the distance to the flower, \(x\).
- Energy gained is therefore \(\propto\) \(q\), cost increases linearly with \(x\).
- Can therefore refactor efficiency as: \(\frac{q}{1+\alpha x} - 1\), where \(\alpha\) is the energy spend per unit of distance.
- Energy spend cannot be negative, so the function must be positive:
\[\sigma(q,x) = [\frac{q}{1+\alpha x}-1]_+\]
Scout distribution
- Assume scouts search an environment without information and report back the first resource found.
- Floral resources should be randomly distributed around a hive and thus follow a Poisson distribution.
- The distance between a central hive and each point should therefore be exponentially distributed, with rate \(\lambda_s\).
- This scout distribution is altered by the profitability function, which translates the distance of the resource to a number of dances reported on the dance floor.
\[\sigma(q, x) \lambda_s e^{-\lambda_s x}\]
Recruit distribution
- Recruits sample randomly from the dance on the dance floor (scouts + other recruits).
- Recruits over-represent the best resources, skewing the distribution of dances.
- The best resource is implicitly a function of distance and should bias towards the closest patches to the colony.
- The chance of discovering a food source within a radius of \(x\) in a circular area is: \(e^{-\lambda_r \pi x^2}\), where \(\lambda_r\) is the scout discovery rate.
- The distribution of the nearest food sources that recruits would come for is: \(2 \lambda_r \pi x e^{-\lambda_r \pi x^2}\)
\[\sigma(q, x) 2 \lambda_r \pi x e^{-\lambda_r \pi x^2}\]
Putting it all together
\[\sigma(q, x) ( p \lambda_s e^{-\lambda_s x} + (1-p) 2 \lambda_r \pi x e^{-\lambda_r \pi x^2})\]
- The distribution of reported distances is a superposition of recruit and scout distance distributions.
- \(p\) provides a proportional weighting of the number of dances coming from either a scout or recruit strategy.
Model of honeybee foraging. Flowers advertised by scouts are distributed exponentially (A), y axis is the log probability of sampling a value larger than \(x\) (same for C and E). These dances are advertised on the dance floor (B) in relation to their profitability, meaning sampling recruits are biased to the more profitable (and closer) resources (C). Recruits also dance for these resources leading to higher recruitment (D) which overall skews the distribution of distances reported on the entire dance floor (E).